the logica yearbook 2000 - scholarly publications

THE LOGICA YEARBOOK

2000

EDITED BY ONDREJ MAJER

FILOSOFIAPRAGUE 2001

Published by FILO50PIA-9IA020'PtA

Institute of Philosophy

Academy of Sciences of the Czech Republic,

Prague

Edited by Ondrej Majer

Design and typesetting by Martin Pokorny

The Logica Yearbook 2000 title and cover © Ondrej Majer

Copyright of the papers held by the individual authors, unless otherwise noted

Printed by PB tisk Pfibram, Czech Republic

The publication of this volume was made possible by research grant #A0009001

'LOGICA' of the Grant Agency of the Academy of Sciences of the Czech Republic

ISBN 80-7007-149-4

TABLE OF CONTENTS

A PREFACE 9Gabriel Sandu, Ahti PietarinenINFORMATIONAL!^ INDEPENDENT CONNECTIVES 11Sten LindströmQUINE'S INTERPRETATION PROBLEM AND THE EARLY DEVELOPMENTOF POSSIBLE WORLDS SEMANTICS 29Michael BeaneyCONCEPTIONS OF LOGICAL ANALYSIS 57Jaroslav PeregrinABSOLUTE AND RELATIVE CONCEPTS IN LOGIC 71Woosuk ParkON COCCHIARELLA'S RETROACTIVE THEORY OF REFERENCE 79Gabriele UsbertiMEANING AND EMPIRICAL JUSTIFICATION 91Margaret CuonzoINTUITION AND PARADOX 101Petr JirkùHow TO UNDERSTAND NEGATIVES I l lTomasz PlacekA PUZZLE ABOUT CONDITIONALIZATION 123Frantisek GaherNEGATION AND PRESUPPOSITION 133Göran SundholmA PLEA FOR LOGICAL ATAVISM 151Marie Duzi, Pavel MaternaPROPOSITIONAL ATTITUDES REVISED 163

Björn JespersenORTCUTT, OEDIPUS AND OTHER DE RE PUZZLES 175Jari PalomäkiTHE SUBJECT MATTER OF MATHEMATICS. TICHY'S VIEW CONSIDERED . 193Ondrej Majer(IN)TRANSPARENT INTENSIONAL LOGIC (SOME REMARKS

TO THE NOTION OF TRIVILISAT1ON IN TIL) 205

A Plea for Logical Atavism"

Göran Sundholm

1. Jean van Heijenoort, and following him, Jaakko Hintikka, have pointed to acluster of distinctions that serve to identify some of the (Collingwoodian)"absolute presuppositions" of 20th century logic.' These distinctions comprise,among others:

LOGIC AS LANGUAGE VERSUS LOGIC AS CALCULUS

(i) Universal language versus many meta-perspectives;(ii) Universal logical laws with content versus meta-linguistic laws

without content.(iii) Formal system for proving theorems in versus formal systems for prov-

ing theorems about,(iv) One fixed universe versus varying domains of discourse (Boole-

Schroder tradition).

My self-chosen office is that of counsel for the defence to Logical Atavism, aposition that is closely connected to the Van Heijenoort-Hintikka theme.

The Oxford fnglish dictionary (first edition) defines:

Atavism, [a. F. atavisme, f. L. aiav-us a great-grandfather's grandfather, anancestor; cf. av-us grandfather.) Resemblance of grandparents or moreremote ancestors rather than to parents; tendency to reproduce the ances-tral type in animals or plants.

Who, then, are the ancestors of the present-day homo iogicusf Answering onlyfor myself, my immediale logical ancestors, that is, my Doktortäler, were meta-logicians.'' The meta-logical paradigm, to which they, as well as the over-

* Text of an invited lecture a! LOGICA 2000. The material was also presented in Paris, March13th. 2000, at an ÎHPST seminar, and I am indebted to J. Dubucs and F. Pataut for their kind invi-tation. There is some overlap with other writings of mine, in particular [1998b] and [forthcomingb]. Dr. Björn Jespersen (Leyden) read my penultimate draft and saved me from a number of splips.1 Van Heijenoort [1967], 11976]. Hintikka[ I9S8], [1996] prefers to recast the distinction in termsof Language as the Universal Medium versus Language as Calculus.2 At Uppsala. Stig Kanger. one of the creators of the Beth-Hmtikka-Kanger-Schütte method of

152 Göran Sundhoim

whelming majority of contemporary logicians, belong, was established onlyafter 1930 as a result of the fundamental contributions offered by Kurt Gôdeland Alfred Tarski, and their masterly codification by Paul Bernays.J Prior tothis metamathematical shift, the foremost task of any "Symbolic Logician"worth his salt was not to prove metamathematical theorems about forma! lan-guages, metamathematically construed, but to provide a foundation for mathe-matics in the following way:

(1) designing a sizeable formal language with an axiomatic deductiveapparaturs;

(2) providing careful meaning-explanations for its basic (or primitive)notions;

(3) making the axioms and rules of inference evident form the meaning-explanations in question,

such that

(4) the resulting framework is adequate (at least in principle) for real andcomplex analysis.4

2. Our Founding Fathers, for sure, exhibit this pattern: in Frege, it is presentwholesale. The Grundgesetze, Vol. I, §§ 29-31, in particular, can be seen asFrege's attempt to establish the adequacy of his meaning explanations.Unfortunately, as we now know with the benefit of hindsight, his splendidefforts fall foul of the Zermelo-Russell paradox. Controversy would haveended, or perhaps better, would not even have begun, if his attempted execu-tion of the above foundationalist programme had been successful. TheGrundlagen crisis would not have taken place. The alternatives offered byBrouwer and Hubert would not have been canvassed and today we would allhave been devout Fregeans.

As it went, a decade after Frege had his life's work shattered by Russell('sparadox), Whitehead and Russell completed the publication of their PrincipiaMathematica, a massive, type-theoretical emendation of Frege's foundationa!scheme in three volumes. Also they adhered to the above foundationalistscheme. However, their attempted execution of the programme is flawed as was

proving completeness by means of applying the Gentzen sequent-calculus rules backwards and ofthe "possible-worlds" approach to semantics for modal logic.At Oxford, Dana Scott, with fundamental contributions to almost all branches of metamathemat-ics. and Robin Gandy, an architect of the theory of the hyperarithmetical and analytical sets.> Oodel [19311. Tarski [1935], Hilbert-Bernays [I93«9|.* This framework, of course, is little but an adaptation to modern needs of the Aristotelian-Euclidean paradigm as set out the Posterior Analytics. Scholz [1930] remains an excellent treat-ment.

A Plea for Logical Atavism 153

speedily realized. Ramsey, in particular, stressed that three of their "axioms"sail under that flag falsely, namely those of Infinity, Reducibility, and Choice(under its Russellian guise as the "Multiplicative Axiom"): the Whitehead-Russell explanations offered in Principia Mathematica simply fail to make evi-dent the "axioms" in question. It is by no means clear that they are principles'eines Beweises weder fähig noch bedürftig'.5 Russell and Whitehead mainlydrew on pragmatic utility-considerations in order to motivate the axiomatic sta-tus of the controversial principles, very much along the same lines that ErnstZermelo had used in order to counter his adversaries in his polemical master-piece |1908a]. Zermelo was clearly aware of the fact the his concomitantaxiomatization of set theory [ 1908b] does not constitute a foundation in thesense of ( 1 ) - (4) above. His conception is postulational in the sense of Hubert,but not Aristotelian-Euclidean in the sense of organized science according tothe Posterior Analytics. Zermelo begins by considering a domain ("Bereich") inwhich the set-theoretical axioms hold and continues from there, after the fash-ion of Hubert's work in geometry that he had witnessed at close range inGöttingen.

Furthermore, as was famously noted by Gödel, from the point of view offormal precision, Principia Mathematica presents 'a considerable step back-wards as compared with Frege'.6 Indeed, owing to Russell's sloppiness in thefundamental syntactic and meaning-explanatory parts of Volume I, it evenproved necessary for Whitehead to add a 'Prefatory Statement of SymbolicConventions' to Volume II.'

L. E. J. Brouwer's intuitionistic critique of classical reasoning in mathe-matics, which began in his thesis [ 1907) and was made fully explicit in his brief[ 1908], can, even though Brouwer was notoriously hostile to language, be seenas offering another approach to the problem of content in the foundations ofmathematics. This line of thought was continued by his pupil Arend Heytingaround 1930. Points {1 ) - (4) are present in the foundational perspective gen-erated by his meaning-explanations, possibly with some reservation about theextent to which he succeeds in realizing (4).s

3. Both the Logicism of Frege, Russell and others, as well as the Intuitionismof Brouwer, unambiguously fall within the foundationalist scheme. Formalismunder the guise of Hubert's Programme, on the other hand, can (best) be seenas an attempt to circumvent the problem of missing content withinFoundations. Here only anodyne propositions {that are acceptable as con-

! Frege i 1884. §3. p. 4].'GSdelil944, p. 126|.' See Lowe 11985. p. 2921.

], [1930al,11930b],

154 Göran Sundholm

strucvtive by Kronecker and Brouwer alike) were officially held to be mean-ingful. The rest of mathematical language was seen as a formalist game withinstrumental use only. Thus viewed Hubert's formalism constitutes an applica-tion of the positivist slogan:

THE VERIFIABLE CONSEQUENCES CHECK OUT.

What, if anything, the higher ("ideal" in Hubert's terminology) mathemat-ical language speaks about does not matter in the least, as long as everythingnumerical that is derivable with its use can actually be derived without suchuse. Finitistically secured consistency turns out to be enough to ensure that anytheorem with finitist content can actually be proved finitistically. In this fash-ion, Hilbert transforms the extra-mathematical problem of a foundation formathematics into a mathematical problem, namely the mathematical problemof proving finitistically that a certain formal system, say, for analysis, is con-sistent. That such radical simplicity would appeal to mathematicians was aforegone conclusion: in place of abstruse metaphysical deliberations, they areoffered a clear-cut mathematical problem to be solved. Again, as in the case ofFrege, there was hope for an Endlösung to the foundationat issues, and yetagain that hope was frustrated by means of a mathematical result.9

But for these well-known "mainstream" activities that fall within the tradi-tional, though perhaps not wholly accurate Logicism/ Intuitionism/ Formalismpattern, other efforts at saving Logicism were made; Carnap*s [1931] can beseen as its last stand. By [ 1934] even he was prepared to jettison content andadopt anodyne formalism. According to his principle of "Logical Tolerance"'there are no morals in logic' and hence anything goes.

The Polish attempts at revision, restriction, or execution of logicist pro-grammes were more promising. Leon Chwistek and Stanislaw Zaremba(whose writings played a crucial role in stimulating the formalist, metamathe-matical researches of Jaques Herbrand) are two important names here.10

However, the prize for the most formidable exponent of foundationalism afterFrege surely has to go to Stanislaw Lesniewski, in whose writings almost neu-rotic levels of formal precision are reached. His penchant for absolute rigourhas not been surpassed throughout the history of logic. This definitive formalwork, which was published just before 1930, never caught on, though, owing tothe contiguous shift towards metamathematical work that was mentionedabove. Other logicians clearly found it more exciting to read and master thenovel meta-logical investigations of Gödel and Tarski rather than torturing

' In the case of Frege by means of the set-theoretical antinomies, and that of Zermelo-Russell inparticular, and in Hubert's case by means of Gödel's Incompleteness Theorems 11931].10 Zaremba [1926] respectively Chwistek [ 1948] (where further references to his work are found).


themselves by adhering to Lesniewski's "directives"." But for a small band ofdevoted followers no one has paid much attention to his grandiloquent con-structions, whereas it is clear that Lesniewski has much to offer concerning thephilosophy of logic. A Lesniewski revival is in my opinion long overdue. Surelyhis work merits also gênerai, non-sectarian interest.

4. Zermelo [ 1930] constitutes a very interesting attempt at a foundation ofmathematics. There Zermelo gives an (almost) isomorphic characterization ofset theory in terms of full second-order logic. Whether this (almost) categori-cal second-order (Hubert) postulate system can be seen also a real(Aristotelian-Euclidean) axiomatization with content is a highly interestingproblem, and one that applies in even higher degree to Dedekind's second-order characterizations of the naturals and real numbers. In other words:

DOES CATEOORICITY OF A SECOND-ORDER POSTULATE SYSTEM CONFER

CONTENT UPON ITS FORMAL LANGUAGE IN SUCH A FASHION THAT ITS

POSTULATES ARE THEREBY TURNED INTO SELF-EVIDENT AXIOMS?

An attempt at an answer must be left for another occasion.Early American attempts at a logicist foundation, incorporating axioms

with content and classical logic, were made by Curry, Church, who used novelformal calculi for the manipulation of functions. These efforts, and the some-what later one by Quine, illustrate some of the dangers that are inherent infoundational work of this kind, where one balances of the edge of the founda-tional abyss. The constructional challenge posed by the foundational scheme isquite difficult, demanding that one gets as much mathematics into the systemas possible with as weak a framework as possible. There is always the risk offalling into the trap of overloading content and falling into inconsistency. Theabove Americans did, for sure, but so did Frege.

By the end of World War II, the metamathematical paradigm was firmlyentrenched. The WFF's of a formal system do not have content but are merelyelements of a freely generated algebra over a suitable alphabet: metamathemat-ical formal languages are not meant for use but for mention only. Sometimes,indeed, the "object language" is left out entirely: it is not used for saying any-thing, but only serves as a subject matter for the (meta)mathematical theorems.12

11 Contributory factors in the neglect were also the facts that his papers are difficult to obtain (andwritten in Polish). His premaiure death just prior to World War II and the ensuing destruction ofmuch unpublished material completed the rout. Perhaps with the appearance of [I992| matters

wilt be remedied ....': Thus, the early presentations of Gödef's Theorem Mostowski 11952] and Feferman [1960]directly work with the Oodel numbers: for instance, a "FREE VARIABLE" is a number, say of the form29+4k, k - 0, 1, 2,..., and a "BOUND VARIABLE" is a number of the form 31+4A, k - 0, 1, 2,...* etc..

156 Göran Sundholrn

A particularly clear example of the shift in perspective and aims is offered bythe changing role of turnstile ' |—*: in early logical works it plays the role of an"assertion sign", indicating that the theorem in question is known while proved.Today, on the other hand, it is universally used a as "derivability" or "theorempredicate" among the WFF's of a formal system.13

5. My atavistic position is now that we as logicians would do well to return notto our immediate ancestors, but to pre-1930 days, when interpreted formal sys-tems still held sway. What, if any, would the advantages of such a retrogradestep be? I list a few.

First, the object language is not longer useless, but its "expressions" willalso express. They are expressions in a real sense and have meaning. Linguisticintuitions concerning grammar and meaning that are used derivatively, so tosay at second hand, in designing the formal semantics of metamathemaitcswould now be used directly, at first hand, for formulating the syntactic struc-ture of the language and for providing the required meaning-explanations. Thecheck on the adequacy of the formalism now rests in its use, rather than in thereasonableness of its metamathematical semantics.

Second, a number of phenomena that tend to get ignored in metamathe-matical treatments are easily taken care of. A prime example is that of asser-toric force (treated by means of the Urieilsstrich, rather than the theorem pred-icate) and other pragmatic notions. When Richard Montague, as distinguisheda representative of the Tarski metalogical school as any, treated of "pragmat-ics" in eponymous papers, what he in fact did was to give a formal semanticsfor demonstrative terms involving reference to a speaker, but nevertheless onlyat the level of content. The use level remains untouched.14

Third, in metamathematical formalism the WFF's play a double role. Incurrent metamathematical treatments of logic, in particular, in systems ofGentzen's Natural Deduction, both formation-rules and derivation-rules per-tain to WFF's.15 Thus, they serve as formalistic simulacra of propositions, thatis, judgmental contents, since according to the syntactic formation-rules theyare the constituents out of which other more complex WFF's are built. On theother hand, they also play the role of demonstrated ("asserted") theorems,since WFF's serve as end-formulae of the derivation-trees. This conflation ofroles has led

Fourth, to a neglect of the distinction between (alethic) logical conse-quence (that is, a relation between propositions) and (epistemic) inference

'J Stepanians [1998, chs. 1-5| tel!s the exciting story of the Fregean assertion sign. KJeene [1952.p. 88, p. 526] documents its use as a theorem predicate." For instance, Montague 11970J.l! In my | I998a) and [2000] I deal with the semantic interpretation of natural deduction deriva-tion trees.


(that is, an act of passage between theorems). For instance, one way to look atthe entailment connective of "relevance logic" is that it propositionalizes thesecondary relation that holds between the propositions that serve as contents of,respectively, the premise-judgement and conclusion-judgement of a {onepremise) valid inference.16

Fifth, also the important semantic notion of presupposition is readilyaccommodated. For instance, definite descriptions are dealt with as follows:

(3!jteind)AU) is true

(ixsInd)A(j) eind B(JT) is a proposition, provided that xelnd

B((ixeInd)A(x)) is a proposition.17

Here we have, contra Wittgenstein's Tractatus 2.0211, an example of howwhether one proposition, namely B((weInd)A(jr)), has sense, does depend onwhether another proposition, namely (3!xelnd)A(;c), is true.

6. Finally, what, if any, atavistic treatments oflogic are there today? To the bestof my knowledge there are only two serious alternatives here. On the classicalside there is TIL, Transparent Intensional Logic, the impressive creation of thelate Pavel Tichy, which has been explored and enriched by the tireless work ofPavel Materna [ 1998] and his Czech, Danish, and Finnish co-workers. Tichy'satavism is cast in the shape of a type theory, using partial functions and classi-cal logic. The resulting type-structure is richer than the standard one, say, ofMontague grammar, owing to the presence of further ground-types and a largenumber of logico-linguistic phenomena have been dealt with successfully. Iwish to deny neither the power of the TIL-machineery nor its flexibility; in my[2000a] I have, however, set out a number of reasons for doubting that it wil!actually run. The basic assumptions of the framework are very strong indeedand, in my opinion, they have not been made sufficiently clear as yet.

That leaves the intuitionistic type theory of Per Martin-Lof [1984J as thesole remaining atavistic alternative known to me. Its linguistic potential is atleast as impressive as that of Tichy's TIL, as has been amply shown in theworks of Aarne Ranta.1* Its versatility in the philosophy oflogic I have attempt-

" I have treated of the conflation between consequence and inference in (1998], j[forthcoming a], where a treatment or the semantic interpretation of natural deduction and of thenotion of assumption can also be found." Stenlund (1973] contains a beautiful treatment of presuppositions along these äines. Ind hereindicates the relevant type of individuals, whereas the exclamation-mark in the existential quantifi-er indicates uniqueness." Ranta 11994] and many subsequent writings.

158 Göran Sundholm

ed to document in treatments of, for example, identity, inference, and the the-ory of truth.19 Martin-Lof himself in a number of works has explored the philo-sophical perspective of his theory within metaphysics and ontology20, episte-mology21 and verifcationism,22 meaning-theory and semantics.23 However, Ilabour under no illusion that these lists, no matter how long or impressive Icould make them, will serve to gain proselytes for logical atavism. The only wayto become convinced of the virtues of logical atavism is to practice it, by exper-imenting with fomal languages having content. The type theory is a system inuse, with meaning. There is no substitute for honest work in order to learn foroneself how thoroughly natural such a system is. Therefore:

ATAVISTS OF ALL NATIONS, UNITE!RETURN TO CONTENT!

Göran SundholmUniversity of LeidenFilosofisch InstituutP.O.Box 9515NL-2300 RA LeidenThe [email protected]

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160 Görnn Sundholm

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162 Coran Sundholm

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Sundholm, B. G., 2000a, 'Virtues and Vices of Interpreted ClassicalFormalisms: Some Impertinent Questions for Pavel Materna on theoccasion of his 70th Birthday', in: Between Words and Worlds. A Festschriftfor Pawel Materna (Timothy Childers and Jari Palomaki, eds.), FilosofïaPubl., Prague, pp. 3-12.

Sundhohn, B. G., forthcoming a, 'A Century of Inference: 1837-1936', in:Jan Wolenski (ed.), Proceedings of IMPS 11, Cracow 1999, Kluwer,Dordrecht.

Sundholm, B. G., forthcoming b, 'Tarski and Lesniewski on Languages withMeaning versus Languages without Use: A 60th Birthday Provocationfor Jan Wolenski', forthcoming in a Festschrift for Jan Wolenski.

Sundholm, B. G., forthcoming c, 'Antirealism and the Roles of Truth', in:Handbook of Epislemology, Kluwer, Dordrecht.

Tarski.Alfred, 1935, 'Der Wahrheitsbegriff in den formalisierten Sprachen',Stadia Phitsophica I, Lemberg, pp. 260-405.

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Paris.Zermelo, Ernst, 1908, 'Untersuchungen über die Grundlagen der

Mengenlehre I', Mathematische Annalen 65, pp. 261-281.Zermelo, Ernst, 1930, 'Über Grenzzahlen und Mengenbereiche: neue

Untersuchungen über die Grundlagen der Mengenlehre', FundamentaMathematkae 16, pp. 29-47.

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